One philosophy of geophysical data analysis
called ``inverse theory''
says that missing data is irrelevant.
According to this philosophy,
a good geophysical model only needs to fit the real data,
not interpolated or extrapolated data,
so why bother with interpolated or extrapolated data?
Even some experienced practitioners belong to this school of thought.
My old friend Boris Zavalishin says,
``Do not trust the data you have not paid for.''

I can justify data interpolation in both human and mathematical terms.
In human terms, the solution to a problem often follows from
the adjoint operator, where
the data space has enough known values.
With a good display of data space,
people often apply the adjoint operator in their minds.
Filling the data space prevents distraction and confusion.
The mathematical justification is that
inversion methods are notorious for slow convergence.
Consider that matrix-inversion costs are proportional
to the cube of the number of unknowns.
Computers balk when the number of unknowns goes above one thousand;
and our images generally have millions.
By extending the operator (which relates the model to the data)
to include missing data,
we can hope for a far more rapid convergence to the solution.
On the extended data, perhaps the adjoint alone will be enough.
Finally, we are not falsely influenced by the ``data not paid for''
if we adjust it so that there is no residual between it and
the final model.